Square trisection
Encyclopedia
The square trisection consist in cutting a square into pieces that can be rearrange to form three identical squares.
partitions
is a geometrical problem that dates back to the Islamic Golden Age
. Craftsman who mastered the art of zellige
need innovative techniques to achieve their fabulous mosaics with complex geometric figures. Thus, the first solution to this problem has been proposed in the 10th century AD by the Persian mathematician Abu'l-Wafa'
(940-998) in his treatise "On the geometric constructions necessary for the artisan". Abu'l-Wafa'
used also his dissection to demonstrate the Pythagoras' theorem
. This geometrical proof of Pythagoras' theorem will be rediscovered in the years 1835 - 1840 by Henry Perigal and published in 1875.
uses 9 pieces. In the 14th century Abu Bakr al-Khalil gives two solutions, one of which uses 8 pieces. In the late 17th century Jacques Ozanam
comes back to this issue and in the 19th century, solutions using 8 and 7 pieces are found, including one given by the mathematician Édouard Lucas
. It's in 1891 that Henry Perigal publish the first known solution with only 6 pieces (see illustration below). Nowadays, new dissections are still found (see illustration above) and the conjecture that 6 is the minimal number of necessary pieces remains unproved.
Problem History
The dissection of a square in three congruentCongruence (geometry)
In geometry, two figures are congruent if they have the same shape and size. This means that either object can be repositioned so as to coincide precisely with the other object...
partitions
Partition of a set
In mathematics, a partition of a set X is a division of X into non-overlapping and non-empty "parts" or "blocks" or "cells" that cover all of X...
is a geometrical problem that dates back to the Islamic Golden Age
Islamic Golden Age
During the Islamic Golden Age philosophers, scientists and engineers of the Islamic world contributed enormously to technology and culture, both by preserving earlier traditions and by adding their own inventions and innovations...
. Craftsman who mastered the art of zellige
Zellige
Zellige, zillij or zellij is terra cotta tilework covered with enamel in the form of chips set into plaster. It is one of the main characteristics of the Moroccan architecture...
need innovative techniques to achieve their fabulous mosaics with complex geometric figures. Thus, the first solution to this problem has been proposed in the 10th century AD by the Persian mathematician Abu'l-Wafa'
Abul Wáfa
Abū al-Wafāʾ, Muḥammad ibn Muḥammad ibn Yaḥyā ibn Ismāʿīl ibn al-ʿAbbās al-Būzjānī was a Persian mathematician and astronomer who worked in Baghdad...
(940-998) in his treatise "On the geometric constructions necessary for the artisan". Abu'l-Wafa'
Abul Wáfa
Abū al-Wafāʾ, Muḥammad ibn Muḥammad ibn Yaḥyā ibn Ismāʿīl ibn al-ʿAbbās al-Būzjānī was a Persian mathematician and astronomer who worked in Baghdad...
used also his dissection to demonstrate the Pythagoras' theorem
Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...
. This geometrical proof of Pythagoras' theorem will be rediscovered in the years 1835 - 1840 by Henry Perigal and published in 1875.
Search of optimality
The beauty of a dissection depends on several parameters. However, it is usual to search for solutions with the minimum of parts. For from being minimal, the square trisection proposed by Abu'l-Wafa'Abul Wáfa
Abū al-Wafāʾ, Muḥammad ibn Muḥammad ibn Yaḥyā ibn Ismāʿīl ibn al-ʿAbbās al-Būzjānī was a Persian mathematician and astronomer who worked in Baghdad...
uses 9 pieces. In the 14th century Abu Bakr al-Khalil gives two solutions, one of which uses 8 pieces. In the late 17th century Jacques Ozanam
Jacques Ozanam
Jacques Ozanam was a French mathematician.-Biography:Jacques Ozanam was born in Sainte-Olive, Ain, France....
comes back to this issue and in the 19th century, solutions using 8 and 7 pieces are found, including one given by the mathematician Édouard Lucas
Edouard Lucas
François Édouard Anatole Lucas was a French mathematician. Lucas is known for his study of the Fibonacci sequence. The related Lucas sequences and Lucas numbers are named after him.-Biography:...
. It's in 1891 that Henry Perigal publish the first known solution with only 6 pieces (see illustration below). Nowadays, new dissections are still found (see illustration above) and the conjecture that 6 is the minimal number of necessary pieces remains unproved.
See also
- Proof by rearrangement of Pythagorean theorem
- Dissection puzzleDissection puzzleA dissection puzzle, also called a transformation puzzle or Richter Puzzle, is a tiling puzzle where a solver is given a set of pieces that can be assembled in different ways to produce two or more distinct geometric shapes. The creation of new dissection puzzles is also considered to be a type of...
- TangramTangramThe tangram is a dissection puzzle consisting of seven flat shapes, called tans, which are put together to form shapes. The objective of the puzzle is to form a specific shape using all seven pieces, which may not overlap...